[Author]
 Oleg I. Reinov

[Title]
 Around Grothendieck's theorem on operators with nuclear adjoins

[AMS Subj-class]
 46B28 Spaces of operators; tensor products; approximation properties

[Abstract]
 In 1955, A. Grothendieck proved that if an operator $T: X\to Y$ in Banach
 spaces has a nuclear adjoint, then $T$ is nuclear provided that $X^*$ has
 the approximation property. It was shown by T. Figiel and W.\,B. Johnson (1973)
 that the assumption on $X^*$ is essential. A generalization of Grothendieck's 
 theorem was obtained in 1987 by E. Oja and O. Reinov: The conclusion of the 
 theorem is true if $Y^{***}$ has the approximation property (and the assumption 
 posed on $Y^{***}$ is also essential). A "revised" (more general) version of 
 the theorem was formulated and proved in 2012 by E. Oja for weak${}^*$ 
 continuous operators from a dual Banach space into an arbitrary Banach space.
 She considered even the cases of $\alpha$-nuclear operators for tensor norms 
 $\alpha.$ We present here, among other results, a "revised" version of 
 Grothendieck's theorem in the cases of $\alpha$-nuclear operators for 
 projective tensor quasi-norms $\alpha.$ This is a talk at the 27th 
 St.Petersburg Summer Meeting in Mathematical Analysis (2018) and also
 a preprint of the paper that will be submitted to the Journal of Mathematical 
 Analysis and Applications.

[Comments]
 LaTeX, English, 24 pp.

[Contact e-mail]
 orein51@mail.ru